This paper studies the recovery guarantees of the models of minimizing ||X||∗ + 1/2a ||X||2F where X is a tensor and ||X||∗ and ||X||F are the trace and Frobenius norm of respectively. We show that they can efficiently recover low-rank tensors. In particular, they enjoy exact guarantees similar to those known for minimizing ||X||∗ under the conditions on the sensing operator such as its null-space property, restricted isometry property, or spherical section property. To recover a low-rank tensor X0, minimizing ||X||∗ + 1/2a ||X||2F returns the same solution as minimizing ||X||∗ almost whenever α ≥ 10max ||X0(i)||2.